Understanding Modulus of Elasticity

When a material is subjected to an external force, it undergoes deformation. If the material returns to its original shape after the force is removed, it is said to be elastic. Within the elastic limit, there is a proportional relationship between the stress applied to the material and the resulting strain. This relationship is described by Hooke's Law.

Let's define the key terms:

• Stress: Stress ($\sigma$) is defined as the internal restoring force acting per unit area of the deformed body. It is a measure of the intensity of the internal forces that particles of a material exert on each other. For longitudinal stress, it is the force applied perpendicular to the cross-sectional area, given by $\sigma = \frac{F}{A}$, where $F$ is the force and $A$ is the area.
• Strain: Strain ($\epsilon$) is defined as the ratio of the change in dimension of the body to its original dimension. It is a measure of the deformation. For longitudinal strain, it is the change in length ($\Delta L$) divided by the original length ($L_0$), given by $\epsilon = \frac{\Delta L}{L_0}$.
• Elastic Limit: This is the maximum stress a material can withstand without undergoing permanent deformation. Within this limit, the material exhibits elastic behavior.

According to Hooke's Law, within the elastic limit, stress is directly proportional to strain: $$ \text{Stress} \propto \text{Strain} $$ The constant of proportionality is known as the modulus of elasticity. There are different types of moduli of elasticity depending on the type of stress and strain involved.

Types of Moduli of Elasticity

The ratio of stress to strain within the elastic limit is generally referred to as a modulus of elasticity. Specific names are given based on the type of deformation:

• Young's Modulus (Y): This modulus relates longitudinal stress to longitudinal strain. It is applicable to solids and measures the stiffness of a material under tension or compression. $$ Y = \frac{\text{Longitudinal Stress}}{\text{Longitudinal Strain}} = \frac{\sigma}{\epsilon} $$ This is often simply called the modulus of elasticity when dealing with longitudinal deformation.
• Shear Modulus (G) or Modulus of Rigidity: This modulus relates shear stress to shear strain. It measures a material's resistance to shearing deformation.
• Bulk Modulus (K): This modulus relates volumetric stress (pressure) to volumetric strain. It measures a material's resistance to compression under uniform pressure.

The question asks for the ratio of longitudinal stress to strain within the elastic limit. Based on the definitions above, this specific ratio is known as Young's Modulus. While 'Young's Modulus' is the precise term, it is a form of 'modulus of elasticity'.

Considering the options provided:

• 'modulus of elasticity': This is a general term that encompasses Young's Modulus, and in the context of longitudinal stress and strain, it refers to Young's Modulus.
• 'shear modulus of elasticity': This relates to shear stress and strain, not longitudinal.
• 'bulk modulus of elasticity': This relates to volumetric stress and strain, not longitudinal.
• 'tangent modulus of elasticity': This term is typically used to describe the slope of the stress-strain curve at any specific point, often beyond the proportional limit where the relationship may not be linear. The question specifies "within elastic limit" where the relationship is linear and constant.

Therefore, the ratio of longitudinal stress to strain within the elastic limit is known as the modulus of elasticity, specifically Young's Modulus.